An Atwood Machine Primer
Below is an Atwood machine which is simply a pulley with a mass at each end of a string passed over the top of the pulley and the free body diagram for each mass.
There are a few assumptions made to simplify the study of an Atwood machine. First, the pulley is considered to be an ideal pulley. An ideal pulley is frictionless and massless. Friction would result in accelerations being smaller than the calculated values. The mass of the pulley would introduce a small amount of rotational inertia which results in the pulley resisting changes in its state of rotation. The string is assumed to be massless and unstretchable. This ensures that the mass of the system consists only of the two masses at each end of the string and the tension is the same throughout the length of the string. Despite these simplifications, the experimental results can be quite good. The reason for using an Atwood machine is to provide a simple way to produce a small acceleration compared to the acceleration due to gravity. The assumptions that have been discussed make the following equations valid: Fnet = m1a (1) Fnet = m2a (1) Fnet = T – Fw1 (2) Fnet = Fw2 – T (2) m1a = T – m1g (3) m2a = m2g – T (3) Equation (1) is Newton’s 2nd law in its general form, eqn (2) is the net force applied to each mass from the free body diagram, and eqn (3) results by equating eqns (1) and (2). Adding both equations (3) cancels the T and rearranging terms yield: a = (m2 – m1)/(m2 + m1) • g where the masses can be expressed in any mass units.